L(s) = 1 | + 2-s − 3.12·3-s + 4-s − 1.12·5-s − 3.12·6-s + 2.48·7-s + 8-s + 6.76·9-s − 1.12·10-s − 6.24·11-s − 3.12·12-s + 2·13-s + 2.48·14-s + 3.51·15-s + 16-s − 17-s + 6.76·18-s + 2.48·19-s − 1.12·20-s − 7.76·21-s − 6.24·22-s + 6.88·23-s − 3.12·24-s − 3.73·25-s + 2·26-s − 11.7·27-s + 2.48·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s − 0.503·5-s − 1.27·6-s + 0.939·7-s + 0.353·8-s + 2.25·9-s − 0.355·10-s − 1.88·11-s − 0.902·12-s + 0.554·13-s + 0.664·14-s + 0.907·15-s + 0.250·16-s − 0.242·17-s + 1.59·18-s + 0.570·19-s − 0.251·20-s − 1.69·21-s − 1.33·22-s + 1.43·23-s − 0.637·24-s − 0.746·25-s + 0.392·26-s − 2.26·27-s + 0.469·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.281740962\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281740962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 - 0.640T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 + 0.484T + 41T^{2} \) |
| 43 | \( 1 - 7.28T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 5.45T + 53T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 6.64T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.299097457128247099959787872900, −7.84650792328682164331781548960, −7.57899746998735347888731477967, −6.60550286409523413965575622974, −5.60042228695587129189553254734, −5.24731526291170477036117291163, −4.62348079704312820967578544449, −3.61409259929070984858652157173, −2.13454295235440367016848618009, −0.74212082551275210617951227077,
0.74212082551275210617951227077, 2.13454295235440367016848618009, 3.61409259929070984858652157173, 4.62348079704312820967578544449, 5.24731526291170477036117291163, 5.60042228695587129189553254734, 6.60550286409523413965575622974, 7.57899746998735347888731477967, 7.84650792328682164331781548960, 9.299097457128247099959787872900